Respuesta :
Answer:
[tex]\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| - \frac{1819}{20}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
- Left to Right
Equality Properties
- Multiplication Property of Equality
- Division Property of Equality
- Addition Property of Equality
- Subtraction Property of Equality
Algebra I
- Functions
- Function Notation
Calculus
Derivatives
Derivative Notation
Antiderivatives - Integrals
Integration Constant C
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle f'(t) = t^4 + 91 - \frac{3}{t}[/tex]
[tex]\displaystyle f(1) = \frac{1}{4}[/tex]
Step 2: Integration
Integrate the derivative to find function.
- [Derivative] Integrate: [tex]\displaystyle \int {f'(t)} \, dt = \int {t^4 + 91 - \frac{3}{t}} \, dt[/tex]
- Simplify: [tex]\displaystyle f(t) = \int {t^4 + 91 - \frac{3}{t}} \, dt[/tex]
- Rewrite [Integration Property - Addition/Subtraction]: [tex]\displaystyle f(t) = \int {t^4} \, dt + \int {91} \, dt - \int {\frac{3}{t}} \, dt[/tex]
- [1st Integral] Integrate [Integral Rule - Reverse Power Rule]: [tex]\displaystyle f(t) = \frac{t^5}{5} + \int {91} \, dt - \int {\frac{3}{t}} \, dt[/tex]
- [2nd Integral] Integrate [Integral Rule - Reverse Power Rule]: [tex]\displaystyle f(t) = \frac{t^5}{5} + 91t - \int {\frac{3}{t}} \, dt[/tex]
- [3rd Integral] Rewrite [Integral Property - Multiplied Constant]: [tex]\displaystyle f(t) = \frac{t^5}{5} + 91t - 3\int {\frac{1}{t}} \, dt[/tex]
- [3rd Integral] Integrate: [tex]\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| + C[/tex]
Our general solution is [tex]\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| + C[/tex].
Step 3: Find Particular Solution
Find Integration Constant C for function using given condition.
- Substitute in condition [Function]: [tex]\displaystyle f(1) = \frac{1^5}{5} + 91(1) - 3ln|1| + C[/tex]
- Substitute in function value: [tex]\displaystyle \frac{1}{4} = \frac{1^5}{5} + 91(1) - 3ln|1| + C[/tex]
- Evaluate exponents: [tex]\displaystyle \frac{1}{4} = \frac{1}{5} + 91(1) - 3ln|1| + C[/tex]
- Evaluate natural log: [tex]\displaystyle \frac{1}{4} = \frac{1}{5} + 91(1) - 3(0) + C[/tex]
- Multiply: [tex]\displaystyle \frac{1}{4} = \frac{1}{5} + 91 - 0 + C[/tex]
- Add: [tex]\displaystyle \frac{1}{4} = \frac{456}{5} - 0 + C[/tex]
- Simplify: [tex]\displaystyle \frac{1}{4} = \frac{456}{5} + C[/tex]
- [Subtraction Property of Equality] Isolate C: [tex]\displaystyle -\frac{1819}{20} = C[/tex]
- Rewrite: [tex]\displaystyle C = -\frac{1819}{20}[/tex]
- Substitute in C [Function]: [tex]\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| - \frac{1819}{20}[/tex]
∴ Our particular solution to the differential equation is [tex]\displaystyle f(t) = \frac{t^5}{5} + 91t - 3ln|t| - \frac{1819}{20}[/tex].
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Integration
Book: College Calculus 10e
